ch 11. quantum states for many-electron atoms and atomic spectroscopy (a.s.) ms310 quantum physical...
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Ch 11. Quantum States for Ch 11. Quantum States for Many-Electron Atoms and Many-Electron Atoms and
Atomic Spectroscopy (A.S.)Atomic Spectroscopy (A.S.)
MS310 Quantum Physical Chemistry
• States of many - e- atoms are grouped into terms and levels
• A.S. is useful for obtaining information on the levels of atoms and
understanding the coupling of and
• A. spectroscopies are widely used in analytical chemistry
• Laser and excited atoms are of interest.
→
S→
l
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11.1 Good quantum numbers, terms, levels,11.1 Good quantum numbers, terms, levels, and statesand states
MS310 Quantum Physical Chemistry
What about the quantum number of many-electron atom?In H atom, n, l, ml, and ms are used.Each operator , , , and commute with the hamiltonian. → independent of time → n, l, ml, and ms : Good quantum number!
However, in many-electron atoms, these numbers are not good quantum number!
Case of Z<40, we can separate the angular momentum and spin momentum. → define total orbital momentum vector L and total spin momentum vector S
2l zl2s zs
)1(||,)1(||,s,l SSSLLLSL ii
i
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MS310 Quantum Physical Chemistry
We define the scalar ML and MS
i
ziSi
ziL sMlM ,
Next, we define , , , and 2L zL2S zS
22,
22, )ˆ(ˆ,ˆˆ,)ˆ(ˆ,ˆˆ
ii
iizz
ii
iizz sSsSlLlL
Good quantum number in many-electron atoms : L, S, ML, MS
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MS310 Quantum Physical Chemistry
Although this configuration is very useful, angular momentum and spin momentum interact in real atom when L>0, S>0 → spin-orbit coupling (a magnetic interaction)
If spin-orbit coupling occurs, operator , , , and don’t commute with hamiltonian.However, the operators and commute with hamiltonian.
Total angular momentum J is defined by
In this case, the only good quantum numbers are J and MJ, the projection of J on the z axis.
2L zL2S zS
2J zJ
SLJ
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
When no electron-electron repulsion : electron ‘configuration’ Take electron-electron repulsion : Term(group of states that has the same L and S) When Z>40, effect of spin-orbit coupling increase, and good quantum numbers are J and MJ : Level(groups of 2J+1 states), energy depends on J. If external magnetic field applied, each state split(same J, different MJ and energy depends on both J and MJ)
ex) Carbon atom configuration 1s22s22p2 : 3 terms, 5 levels, and 15 states
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11.2 The energy of a configuration depends onboth orbital and spin angular momentum
MS310 Quantum Physical Chemistry
Consider the He, 1s12s1 configuration2 electrons are both l=0(s orbital) : |L|=0Magnitude of spin angular momentum vector :Vector s has a 2s+1=2 orientations.In this case, there are 2 spins can only. parallel (α(1)α(2), β(1)β(2)) and antiparallel (α(1)β(2), β(1)α(2))
)1(|s| ss
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MS310 Quantum Physical Chemistry
Calculate the MS value. twice MS = ms1+ms2=0, each MS = ms1+ms2=1 and MS = ms1+ms2=-1
We know S ≥ |MS| → S=1 when |MS|=1, MS=±1 MS takes -S to S : S=1 group include MS = 1,0,-1 → tripletWhen S=0, there are only MS = 0 → singletSinglet and triplet : associated with paired and unpaired electrons
Singlet and triplet wavefunction is given by
}
)]2()1()2()1([2
1)2()1(
)2()1(
)]{2(1)1(2)2(2)1(1[2
1
)]2()1()2()1([2
1)]2(1)1(2)2(2)1(1[
2
1
ssss
ssss
triplet
singlet
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MS310 Quantum Physical Chemistry
Vector model of the singlet and triplet states
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MS310 Quantum Physical Chemistry
We approximate the potential is spherically symmetry.However, if l>0, probability distribution is not spherically symmetrical. → there are different repulsive interaction depending on ml values. → repulsive interaction between electrons : depends on l and s.
Only ‘partially’ filled subshells contribute to L and S.How can one calculate it?
If spin-orbit coupling is neglected : total energy independent from ML and MS. → group of different quantum state : same L and S value, different ML and MS values.(it means degeneracy)
Group of states : ‘term’L and S values for the term : 2S+1L, L=0,1,2,3… : symbol S,P,D,F…
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MS310 Quantum Physical Chemistry
Degeneracy : 2L+1 for L value, 2S+1 for S value → (2L+1)(2S+1) : degeneracy of a term, 2S+1 : multiplicity
If filled subshell or shell, → degeneracy=1(only the 1S)
Term symbol : independent from principal quantum numberC : 1s22s22p2 ,Si : 1s22s22p63s23p2 : same set of terms
How are terms generated for a given configuration? → consider the ‘not-filled’ subshells(filled subshells doesn’t contribute the term)
Possible values of L and S : Clebsch-Gordon seriesFor 2-electron case, allowed L values are given by l1+l2, l1+l2-1, …, |l1-l2| and allowed S values are s1+s2 and s1-s2
0,0 i
siSi
liL mMmM
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MS310 Quantum Physical Chemistry
The different ways in which 2 electrons can be placed in p orbital is shown.
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
Rule 1 : The lowest energy term is that which has the greatest spin multiplicity. For example, the 3P term of an np2 configuration is lower in energy than the 1D and 1S terms.
Relative energy of different terms : the Hund’s rule
Rule 2 : For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy. For example, the 1D term of an np2 configuration is lower in energy than the 1S term.
Hund’s rules imply that the energetic consequences of e- - e-
repulsion are greater for spin than for orbital angular momentum.
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11.3 Spin-orbit coupling breaks up a term 11.3 Spin-orbit coupling breaks up a term into levelsinto levels
MS310 Quantum Physical Chemistry
Until now, we said the all states in a term have the same energy.However, in real case, spin-orbit coupling occurs and terms are split into closely spaced levels.
See the total angular momentum vector JMagnitude of J can take the L+S, L+S-1, …, |L-S| For example, 3P term has J=2,1,0.→ 5 states with 3P2, 3 states with 3P1, 1 state with 3P0
Therefore, total states are 9.
Nomenclature : 2S+1LJ
2J+1 states have different MJ values associated with each J values.Generally, there are (2L+1)(2S+1) states in 2S+1LJ
Coupling : add L•S term into total energy operator
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MS310 Quantum Physical Chemistry
Taking spin-orbit coupling, it gives Hund’s third rule
Rule 3 : The order in energy of levels in a term is given by the following :
If the unfilled subshell is exactly or more than half fill, the level with the highest J value has the lowest energy.
If the unfilled subshell is less than half fill, the level with the lowest J value has the lowest energy.
Use it, we can determine the lowest energy level in same term
Lowest energy of np2 configuration : 3P0 levelLowest energy of np4 configuration : 3P2 level, it describes O
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MS310 Quantum Physical Chemistry
Level diagram of Carbon(ground state)np2 configuration : 3P0 level is lowest level
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Ex) excited configuration of C : 1s2 2s2 2p1 3d1
0,1
1,2,3
,2,1
111
333
21
2121
S
L
ssll
111
03
13
233
211
13
23
33
3
311
23
33
43
3
P101,),01(P0,1
P,P,P0,1,211,),11(P1,1 3)
D202,),02(D0,2
D,D,D1,2,31,),12(,),(
D1,2 2)
F303,),03(F0,3
F,F,F2,3,42,),13(,),(
F1,3 1)
JSL
JSL
JSL
SLSLJ
SL
JSL
SLSLJ
SL
Degeneracy : total 60 states3d electron : 5 different ml, 2 different ms : total 10 combination2p electron : 3 different ml, 2 different ms : total 6 combination → 6x10=60 quantum states in 1s22s22p13d1MS310 Quantum Physical
Chemistry
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MS310 Quantum Physical Chemistry
11.4 The essentials of atomic spectroscopy11.4 The essentials of atomic spectroscopy
Spectroscopy : see the ‘transition’
What transitions are allowed? ‘selection rule’Selection rule : obtained by dipole approximation(8.4)Very useful although forbidden transition in the dipole approximation can occur in higher level theory
Dipole selection rule ∆n=±1 for vibration ∆J=±1 for rotation(J:rotational quantum number)
In atomic level : consider the spin-orbit coupling : ∆l=±1, ∆L=0,±1, ∆J=0,±1 and ∆S=0 (J:total angular momentum, L+S)
How use it? Transition of Cs → ‘atomic clock’(frequency of transition : 9192631770 s-1)
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MS310 Quantum Physical Chemistry
Energy level of H atom :
Absorption frequency is given by
RH : Rydberg constant, 109677.581 cm-1
ninitial=1 : Lyman series ninitial=2 : Balmer series ninitial=3 : Paschen series ninitial=4 : Brackett series ninitial=5 : Pfund series
2220
4
8 nh
emE e
n
)11
()11
(8
~222232
0
4
finalinitialH
finalinitial
e
nnR
nnch
em
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MS310 Quantum Physical Chemistry
More general display : Grotrian diagram
Case of He atomSolid line : allowed, dashed line : forbidden transition
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MS310 Quantum Physical Chemistry
11.5 Analytical techniques based on atomic 11.5 Analytical techniques based on atomic spectroscopyspectroscopy
Example : detect toxic metal using the atomic emission and atomic absorption spectroscopy
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MS310 Quantum Physical Chemistry
Sample : very small droplet(1-10μm)Heated zone : electrically heated graphite furnace or plasma arc source → convert the state to excite states
Atomic emission spectroscopy
Light emitted by excited-state atoms → transitions back down to the ground state : dispersed into its component wavelengths by a monochromator
Intensity : proportional to # of excited-state atoms : character of ‘atom’ nupper/ nlower : 6x10-4 for 3000K Na
Use photomultiplier, spectral transition for nupper/ nlower < 10-10
It used for detect the 589.0nm and 589.6nm Na emission
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MS310 Quantum Physical Chemistry
Atomic absorption spectroscopy Difference : light pass through the heated zone, absorption occurs from lower state to excited states and detected → we see the ‘absorption’
Sensitivity : 10-4 μg/ml for Mg, 10-2 μg/ml for Pt
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11.6 The Doppler effect11.6 The Doppler effect
MS310 Quantum Physical Chemistry
Doppler effect : shift of frequency
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MS310 Quantum Physical Chemistry
Shifted frequency is given by
c
vc
v
z
z
1
1
0
In non-relativistic region, formula is more simple.
c
vz1
10
In real case, ‘distribution’ of speed : follows the Maxwell-Boltzmann distribution. → all velocity directions : ‘equally’ distributed → large range and <vz>=0
Therefore, there are no shift but ‘broadening’ occurs. : Doppler broadening
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11.7 The He-Ne laser11.7 The He-Ne laser
MS310 Quantum Physical Chemistry
Selection rule : ∆l=±1 for electron, ∆L=0 or ±1 for atom
Photon-assisted transition (see 8.2)
- Absorption : photon induces a transition to higher level
- Spontaneous emission : excited state relaxes to lower level
- Stimulated emission : photon induces a transition from
excited state to lower level
221221112 )()( NANBNB System is described by
Use blackbody spectral density function, we can obtain
3
32
21
212112
16,
cB
ABB
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MS310 Quantum Physical Chemistry
If stimulated emission dominant : N2>N1 : population inversionKey of laser : stable population inversion
1 to 4 : external source(electric field)4 to 3 : relaxation(spontaneous emission)2 to 1 : similar to 4 to 3 (spontaneous emission)
Lasing transition : 3 to 2
How can make it? ‘optical resonator’
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
Condition of constructive interference : nλ = n(c/ν) = 2dNext constructive condition : n → n+1Difference of frequency : ∆ν = c/2d, bandwidth of cavity
# of nodes : determined by 2 factors 1) frequency of resonator modes 2) width in frequency of the stimulated emission transition
Width of transition : given by the Doppler broadening(by the thermal motion of gas-phase atoms or molecules)
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MS310 Quantum Physical Chemistry
a) resonator transition : depends on the Doppler linewidth b) through the threshold : only 2 peaks survive
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MS310 Quantum Physical Chemistry
How can He-Ne laser act?
1) He 1s2 configuration(1S term) → 1s2s configuration(1S and 3S term) by the electric field : ‘pumping transition’
2) by the collision, energy of He transfer to Ne(not obey the selection rule) : 1S to 2p55s, 3S to 2p54s
3) by the lasing transition(stimulated emission), these states go to 2p54p and 2p53p : 632.8nm
4) by the spontaneous emission, 2p53p goes to 2p53s
5) by the coalitional deactivation, 2p53s goes to 2p6 (ground state)
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
11.8 Laser isotope separation11.8 Laser isotope separationSpeed of gas : proportional to M-1/2
→ it used for separation of 235U,
material of nuclear bomb
Potential difference occurs the laser
isotope separation. Why?
→ real nuclear potential is not a
coulomb potential : energy difference
(calculated by the perturbation theory)
Difference of IE of U : 2 x 10-3%
It can negligible when l>0 : effective
potential is repulsive potential
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11.9 Auger electron and X-ray photoelectron11.9 Auger electron and X-ray photoelectron spectroscopiesspectroscopies
MS310 Quantum Physical Chemistry
Application of spectroscopy : analysis of gas-phase and surface
Character of these 2 spectroscopies : ejection of electron, measure the electron energy
Electron eject to the material, it through the vacuum and outside the vacuum, electron collide to other materials and loss the energy. → energy of atomic level when electron within the ‘inelastic mean free path’(mean free path : average length of atom and molecule can move without the collision)
Inelastic mean free path 2 atomic layer when 40eV 10 atomic layer when 1000eV
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MS310 Quantum Physical Chemistry
Auger electron spectroscopy(AES) : apply the X-ray and measure the low-energy electron
Principle of AES
1) Inject large energy(electron of X-ray photon)2) An electron is ejected from a low-lying level3) Hole of core electron filled through the relaxation from a higher level electron4) By energy conservation, third electron eject from the higher level
Electron beam is focused to a spot size on the order of 10-100 nm → can make a map of elemental distribution at the solid surface with very high lateral resolution
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MS310 Quantum Physical Chemistry
Schematic diagram of AES
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
X-ray photoelectron spectroscopy(XPS) : apply the X-ray and measure the high-energy electron
By the energy conservation, Ekinetic = hν - Ebinding
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MS310 Quantum Physical Chemistry
We can see the chemical shift in the XPS.
Case of CF3COOCH2CH3
High electronegativity of F → net electron withdrawal to F in the CF3 group → electron in C are deshielding and binding energy increase : large, positive chemical shift
Similarly, C in COO group are large deshielding, too.
C in CH2 group : next to O by the single bond : small chemical shift
C in CH3 group : no electron withdrawal by the any atoms : no chemical shift
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MS310 Quantum Physical Chemistry
Surface sensitivity of XPS : Fe film on a crystalline MgO surface
By the X-ray, 2p electron eject to the Fe surface.Spin angular momentum s → coupled with orbital angular momentum l.
Total angular momentum j with 2 possible values J = L+S, L+S-1, …, |L-S| → j = 1+1/2 = 3/2 and j = 1-1/2 = ½
Ratio of photoemission signal : ratio of degeneracy
21
2
12
12
32
)2(
)2(
2/1
2/3
pI
pI
Therefore, different ratio between Fe(II) and Fe(III)
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MS310 Quantum Physical Chemistry
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MS310 Quantum Physical Chemistry
11.10 Selective chemistry of excited state11.10 Selective chemistry of excited state : O(: O(33P) and O(P) and O(11D)D)
Interaction of sunlight with molecules in the atmosphere : set of chemical reactions
Oxygen : major species in atmosphere reaction dynamic equilibrium of oxygen and ozone.
23
23
32
2
2OOO
OOhO
MOMOO
OOhO
M : other molecule(oxygen or nitrogen)
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MS310 Quantum Physical Chemistry
Less than 315nm, dissociation of oxygen occurs as
)()( 132 DOPOO
1D term : 190kJ/mol of excess energy than 3P termIt is used to overcome an activation barrier to reaction.
molkJQOHOHOHDO
molkJQOHOHOHPO
/120,)(
/70,)(
21
23
1D to 3P transition : ∆S=0 → forbidden : 1D are long-lived species and it depleted by the reaction.
Because of the excess energy of 1D species, it can make the reactive hydroxyl and methyl radical.
341
21
)(
)(
CHOHCHDO
OHOHOHDO
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11.11 Configurations with paired and unpaired 11.11 Configurations with paired and unpaired electron spins differ in energyelectron spins differ in energy
MS310 Quantum Physical Chemistry
Consider the first excited state of He : 1s12s1
Schrödinger equation is given by
Singlet wavefunction is given by(ignore the spin part)
)2,1()2,1()4
ˆˆ(120
2
21 singletE
r
eHH
)]2(1)1(2)2(2)1(1[2
1sssssinglet
21120
2
21 )]2(1)1(2)2(2)1(1)[4
ˆˆ)](2(1)1(2)2(2)1(1[2
1
ddssssr
eHHssssEsinglet
002
2
21 2)2,1(ˆ)2,1(
an
eEddH nsn
Use (H-like orbitals)
21120
2
21 )]2(1)1(2)2(2)1(1)[4
)](2(1)1(2)2(2)1(1[2
1
ddssssr
essssEEE sssinglet
Therefore, integral becomes to
Write the integral as the Esinglet = E1s + E2s + J12 + K12
21120
2
12212
12
2
0
2
12 )]1(2)2(1[)1
)](2(2)1(1[8
,)]2(2[)1
()]1(1[8
ddssr
sse
Kddsr
se
J
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MS310 Quantum Physical Chemistry
In triplet state, result changes to Etriplet = E1s + E2s + J12 – K12
It gives the important result. 1) absence the electron repulsion : Etotal = E1s + E2s
2) including the coulomb repulsion, singlet and triplet state separately, change of the value of energy J12 + K12 and J12 – K12
3) J12 >0, K12>0 : triplet state must be lower energy than singlet stateJ12 : coulomb integral, K12 : exchange integral
Consider electron 2 approaches to electron 1Singlet :
Triplet :
Therefore, triplet wavefunction has a greater degree of electron correlation than singlet wavefunction → unpaired spin has lower energy than paired spin.
)1(2)1(12)]1(1)1(2)1(2)1(1[2
1)]2(1)1(2)2(2)1(1[
2
1ssssssssss
0)]1(1)1(2)1(2)1(1[2
1)]2(1)1(2)2(2)1(1[
2
1 ssssssss
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MS310 Quantum Physical Chemistry
- Study the atomic emission spectroscopy and atomic absorption spectroscopy
- Laser : population inversion(stable excited state)
- Auger spectroscopy : third electron is ejected through intermediate process from incident high energy
- X-ray photoemission spectroscopy :
Summary Summary
Ek=hνincident-Ebinding